Macromolecules 1995,28, 4851-4860
4861
Birefringence of a Polystyrene Solution in Elongational Flow: Effects of Molecular Weight and Solvent Quality Tuan Q. Nguyen,* Guozhu Yu, and HenningH. Kausch Department of Materials Science, Polymer Laboratory, Swiss Federal Znstitute of Technology, MX-D, CH-1015 Lausanne, Switzerland Received June 17, 1994; Revised Manuscript Received March 13, 1995@ ABSTMCT: Elongational flow was created by forcing dilute PS solutions across two jets in exact opposition. The retardation signal was mapped across the birefringence domain as a function of fluid strain rate, polymer concentration, MW, and solvent quality. The use of a fast polarization-modulation technique in conjunction with signal averaging increased the detection limit by over 1order of magnitude in comparison to static-polarization experiments. Retardation could be determined at polymer concentrations down to the ppm level. The critical strain rate (k,) for molecular coil orientation was corrected for polymer dispersity by combining the birefringence curves with the MWD obtained by gel permeation chromatography. In contrast with previous investigations, it was found that 6 , depended on the solvent quality and scaled with polymer molecular weight (M) as WM-’.~in a poor solvent (decalin) and as -M-l,s in good solvents (toluene and l-methylnaphthalene). The possibility of using flow birefringence to determine polymer molecular weight distribution has been assessed and compared to the more conventional technique of gel permeation chromatography. The large dispersion in the birefringence curves suggested a gradual increase in segmental orientation with the fluid strain rate instead of a n abrupt coil-to-stretch transition.
Introduction In the late 19709,Pope and Keller observed a highly localized birefringence zone when dilute polymer solutions were pumped across two narrow orifices in exact opposition. The birefringence was visible only above a critical value of the fluid strain rate (ic)which was on the order of the reciprocal chain relaxation time ( t l ) &c
= BIT,
with B
-
1
(1)
€c
- lit, -
K3”
(3)
In this equation, v is a coefficient equal to 0.50 in a @-solvent and 0.59 in a good solvent.8 In a detailed series of investigations, Keller and Odel19J0established that the dependence of &c on is universal, regardless of the solvent quality. This result implies that the parameter B is not constant but increases with polymer molecular weight as -iVP3 in a good solvent. It was suggested that this “anomaly”may be rationalized if the molecular relaxation time and the critical strain rate obey different scaling 1aws.ll Using a Flory’s type mean-field approach, Rabin derived the following relation for the critical strain-rate instead of eq 3:11,12
From the strain-rate dependence and the amplitude of the birefringence signal, it was concludedl that the chains became highly extended above the critical strain rate ic, in accordance with the abrupt coil-to-stretch transition predicted by de Gennes.2 However, a few points remain unresolved within this &c l/z, M-(’+”) seemingly coherent picture. There is no general agree(4) ment about the exact value for the factor B in eq 1.Part The exponent range of -1.5 to -1.6 predicted by eq 4 of this discrepancy may be traced back to the different was in better agreement with the precedently reported models used to derive eq 1and to the diverse numerical experimental r e s u l t ~ . ~ J ~ expressions for TI. For a Hookean dumbbell, unbound The birefringence equipments used in previous inexpansion in steady elongational flow is predicted at i, vestigations were of the static-type based on the Senar= 0.5/2H( t = ~ dumbbell relaxation time).3 More elabomont compensator. In recent years, several improverate calculations based on the bead-spring model with ments have been made which allow significant gain in hydrodynamic interactions give B = 0.5035, which is and in acquisition time. The class of polarjust 7% above the value for the Hookean d ~ m b b e l l . ~ ~sensitivity ~ imeters that has been most extensively investigated Higher estimations for B are, however, frequently found relies on the modulation of light polarization. In the in the literature, with 1 being the most cited value.6 present paper, a version of the rotary modulation For a non-free-drainingchain, the longest relaxation technique developed by Fuller13 and commercialized by time is usually related to the unperturbed root-meanRheometrics U.S. (Rheometrics Optical Analyzer or square end-to-end distance R through the r e l a t i ~ n : ~ ROA) was used for the measurements. The present work is intended to be purely experimenz1= 0.398R3~$kBT (2) tal with a 3-fold purpose: first, to test the gain in sensitivity attainable with the new generation of polarwhere vs is the solvent viscosity, Jtg the Boltzmann ization-modulated instruments under dynamic condiconstant, and T the absolute temperature. tions of flow; second, to investigate in a more quantiFor eq 1 to be valid, &c should scale with polymer MW tative way the relationship between the critical strain as 1/R3, i.e. rate for birefringence and the chain relaxation time, after due consideration of sample polydispersity; third, to check €or the feasibility of using flow birefringence * To whom correspondence should be addressed. Abstract published in Advance ACS Abstracts, June 15, 1995. as a reliable method of molecular weight characteriza-
- -
@
0024-929719512228-4851$09.00/0 0 1995 American Chemical Society
4852 Nguyen e t al.
Macromolecules, Vol. 28, No. 14, 1995
tion in comparison with more conventional techniques such as gel permeation chromatography.
Experimental Section Chemicals. The polystyrene samples were narrow molecular weight standards from Polymer Laboratories (Shropshire, U.K.) with Mp(MW a t peak maximum) in the rgnge of 1.46 x 106-20 x lo6. The polydispersity index, MwIMn,was ‘1.05 for molecular weights below 4 x lo6 and up to 1.20 for the highest molecular weight samples. The solvents, from Fluka AG (Switzerland),were purified by distillation prior to the experiments. Decalin is a mixture of 58% trans- and 41% cis-decahydronaphthalene 1% tetrahydronaphthalene as determined by gas chromatography. The experimental @temperature for PS.in decalin is 14.8 ‘C.14 The polymer solutions are gently stirred for 24 h and then passed through a 14;um sintered-glass filter prior to experiments.
+
Intrinsic Viscosity of Polystyrene Solutions. Two alternative techniques, depending on the polymer MW, have been employed for the determination of the intrinsic viscosity. This experimental procedure was dictated by the following considerations: it is well-known that dilute polymer solutions are shear-thinning in capillary flow above a wall shear rate p UTI. For high-MW polymers, it is necessary to conduct experiments a t very low shear rate to avoid viscoelastic corrections. In addition, according to our experience, the MW averages quoted by the vendors are only approximate in the ultrahigh-MW range. For example, a PS sample commercialized as a 14.4 x lo6 stcndard has in reality a Mp of 21 x lo6, a M , of 18 x lo6, and M, = 15 x lo6 as determined by GPC, viscometry and low-angle laser light scattering, respectively. The kinematic viscosity of PS solutions with MW 5 3 x lo6 was measured with an automatic Ubbelohde viscometer (Schott AVS-310). After correction for the solvent density (es), the following Mark-Houwink-Sakurada (MHS) relationships have been obtained at 295 K which was also the experimental temperature:
-
PS/decalin
vs = 2.39 mPws [VI = 2.34 x
[email protected]’
e, = 879.8 kgm-3
(5)
PWtoluene
vs = 0.585 mPws
‘w = 2bdL = 4Q/(nd2e)
[17] = 1.58 x
[email protected] e, = 863.1 kgm-3
(6)
PW1-methylnaphthalene
7, = 3.02 mPa.s
[VI = 1.87 x
10-3@70 m3*kg
e, = 1020
generally accurate t o within &lo%. Exceptions were the previously mentioned, “14.4 x lo6”,polymer and a ‘:20 x lo6” standard which had a correct Mp = 19 x lo6 but a M,= 15 x lo6 and a M , = 14 x lo6well below the sample specifications. Molecular Weight Distribution. The molecular weight distribution was determined by gel permeation chromatography (GPC) on a Waters 150CV, equipped with special columns for ultrabigh-MW polymers (TSK G7000HXL and Ultrastyragel lo6 A) and diode-array detection (Kontron Model DAD440). A PC-compatible personal computer was used for data acquisition and analysis. Polymers with MW above 3 x lo6 are particularly prone to mechanochemical degradation. To preserve polymer integrity, extreme care should be exercised during the preparation, injection, and analysis of the samples. Experimental conditions for a successful GPC characterization in the ultrahigh-MW range have been described earlier.l5 ElongationalFlow. The experimental setup employed for the creation of stagnant elongational flow is shown schematically in Figure 1. In the design of the opposed jets cell, interchangeable nozzles were used with a gap distance continuously adjustable with a rotary screw. The jet tips were made of synthetic sapphire with a precisely bored circular orifice connected to a 60” tapered channel. The susceptibility of high-MW polymers to shear-induced degradation imposed severe restrictions on the choice of the pumping system which, in addition, must deliver a constant and pulseless flow rate. After several trials, a 500-mL syringe pump capable of transferring up t o 205 m u m i n of polymer solution (ISCO Model 500D) was finally selected for the experiments. With 0.50 mm nozzle diameters and a gap distance of 0.60 mm, a maximum apparent strain rate of 29 000 s-l was achieved in the present equipment. Higher strain rates could be obtained by either decreasing the orifice entrances or by placing several pumps in parallel.16 Using a smaller jet diameter ensured a stable flow field to the detriment of signal sensitivity following a reduction in the birefringence zone. With the second option, it was important that flow be kept well within the laminar regime. Strain-Rate Calculation. Although the opposed-jets geometry has been thoroughly experimented during the last 2 decades,lJOthe corresponding flow-field modelization is only available recently in the Newtonian limit.17-19 In a narrow region along the symmetry axis, an approximate uniaxial extensional flow is created as fluid is pumped simultaneously into the nozzles. The fluid velocity starts from zero at the stagnation point, reaching uo at the orifice entrance. Assuming a uniform jet entrance velocity, an “apparent”fluid strain rate can be estimated from the relation:
-’ (7)
(v,, solvent viscosity, [VI, intrinsic viscosity; in the above equations, the polymer molecular weight (M) is given in units of kgmol-l). In the MW range ’3 x lo6, the viscosity detector of the Waters 150CV was used for intrinsic viscosity determination. The standard capillary was replaced with 100 cm stainless steel tubing of 0.1 cm internal diameter. By measuring the viscosity of a 19 x I O 6 PS solution as a function of flow rate, it was shown that shear thinning was negligible for flow rates 50.50 mL/min (which correspond to wall shear rates 580 s-l). With the appropriate MHS coefficients and the GPC data, it was verified that the M W s given by the manufacturer were
where DO is the volumetric average fluid velocity a t the orifice, Q the total flow rate, d the jet diameter, and e the gap separating the two nozzles. Most of the previous experiments in opposed-jets flow assumed uniform uniaxial extension in the region close to the central streamline. Recent flow simulations showed that this is not the case.18 Changes in the local extensional strain rate can be as high as 100% over the region defined by the two jet entrances. Fortunately, the birefringence zone is highly localized along the centerline. The radial variation in strain rate over the effective zone probed by the laser beam is limited to < 10%. The use of a locally averaged fluid strain rate which is a function only of the axial or flow direction ( x ) is, therefore, justified: ‘(XI
= C(x)€app
The factor C depends on the exact flow geometry. Under the prevalent flow conditions, C has a value which increases continuously from 0.7 a t the stagnation point to 0.9 near the orifice entrances. To be consistent with previously published results, we will continue to use &app in the present investigation. Since most of the reported data pertain to the stagnation point ( x = 01,
Birefringence of a PS Solution in Elongational Flow 4853
Macromolecules, Vol. 28,No. 14, 1995
Figure 1. Schematic diagram of the flow birefringence system showing the major control and data acquisition components.
true values for 1 can be obtained after multiplying correction factar 0.7.
by the
Results and Discussion Flow Birefringence Measurements. All the experiments have been conducted at room temperature (22 f 0.5 "C). The working principles of polarization-modulated optical rheometers have been reviewed by Fuller.20 The configuration of the optical train to measure flow birefringence is depicted schematically in Figure 2. The intensity and polarization of the light beam can be calculated at any position along the optical train by multiplying the Stokes vector of the incident light with the appropriate Mueller matrices. By using that procedure, the output intensity at the deteetor position can be readily computed as:
r o l l l q hall-wave plate ( ol )
I condensing lens
Z = (Z,42)[1+ R,sin(4wt) + R, cos(4wt)l (10) where RI and Rz are coefficients related to the retardation 6' and the orientation angle a of the birefringence zone with respect to the flow direction:
R,= -cos(2a)
sin(&)
R, = sin(2a) sin(&)
(Ilb)
Experimentally, the dc component (I&)is obtained h m a low-pass filter which is part of the ROA signal conditioner. The coefficients R I and RZ are detected with a dual-phase digital lock-in amplifier (Stanford Research Model 850). The orientation angle, a, is an important experimental parameter which depends on the local flow field. In a general situation like in simple shear flowP1or in single-jet flow,Z2a is a function of the axial and radial positions. The situation is simpler in stagnant elongational flow with all the polymer chains oriented in the same direction at a constant angle a. The retardation 6' can thus be determined from a single evaluation of either R I or RP. One definite benefit of the polarization-modulation technique resides in the fact that the electric vector samples cyclically different projections of the refractive index tensor in the plane normal to the direction of light propagation. This feature renders possible real-time determination of 6' and a in a single experiment, in
_I_ I
I quarter-wave plate ( 07
cb I I I
I
pholcdeleclw
e
Figure 2. Diagram of the flow birefringence experiment showing the arrangement of optical elements. addition to the absolute values of 6' without any additional calibration (for example, 6' is negative for PS as a result of lateral orientation of the phenyl groups). Since the incident intensity is continuously monitored, fluctuations in the laser source are automatically cancelled out, resulting in an enhanced signal-to-noise ratio. Besides the gain in sensitivity, the polarization-modulation technique does not require dark-room facilities
4854 Nguyen et al. 3.5E-3
-A
3.OE-3
-
Macromolecules, Vol. 28, No. 14, 1995
-
I
Me.naphthalene
2.5E.3
1
- - - - 5toluene ---
-
-*-
i
decalin
-
3-
0003
r
2.OE-3
0 002
p
0
1.OE.3
5000
loo00
15000
20000
25000
30000
apparent strain-rate (1.’)
It
/;I
5.OE-4
-250
-200 .150
.lo0
.SO
0
50
Figure 4. Maximum retardation (6’0) plotted as a function of Gap, for a PS mixture (90 ppm M , = 6.85 x lo6 + 90 ppm Mp = 4.34 x lo6): (0) in decalin; (W)in 1-methylnaphthalene.
100
150
200
250
Y (Pm)
Figure 3. Retardation profiles for a 100 ppm PS solution (M, = 6.85 x lo6)recorded across the stagnation point in different solvents: (+) in decalin at Gap, = 25 500 s-l; (0)in toluene at GaPf = 25 500 s-l; (m)in 1-methylnaphthaleneat gap,, = 17 000 s-
.
which has hampered the widespread use of flow birefringence. Flow Birefringence Profiles. The motorized X-Y stage (Micro-ContrBleMM2000) allowed raster scanning simultaneously with “on the fly”, data acquisition under the control of a PC-compatible computer. The “focal point” of the scanning beam was adjusted to coincide with the center of the flow. From diffraction theory, the focal region is approximately cylindrical with a “waist” given by:
d, = 4AflnD
(12)
where il = laser wavelength (632.8 nm), f = focal distance of the lens (63 mm), and D = diameter of the laser beam (0.8 mm at half-width). The effective length, If, of the focal cylinder is approximatively given by:
I,= 16ilf3/nD2
(13)
By blocking and translating the laser light across a pinhole, a value of df = 0.06 mm was found for the diameter of the focused beam, in good agreement with the prediction of eq 12. The axial retardation profiles, scanned along the direction y perpendicular to the flow direction, are given in Figure 3 for the three different solvents. The downward curvature of the baseline in l-methylnaphthalene originates from the solvent birefringence which gives a positive retardation signal. The birefringence, A n ’, is related to the retardation 6‘ by the relation:
(the same notation as given in the literature is used, i.e., a prime for birefringence and a double prime for dichroi‘smZ0).
In eq 14, A is the laser wavelength (632.8 nm), and L , the thickness of the birefringence zone. Due to the inhomogeneous nature of the flow, the birefringence signal is not constant over the scanned volume but is an axisymmetric function of the radial distance r . Cathey and Fuller have shown that the birefringence A n ’(r)can be recovered from the retardation profile, d’Cy), by the inverse Abel t r a n ~ f o r m a t i o n . ~ ~ For dilute PS solutions, all the recorded transversal retardation profiles could be fitted to a Gaussian function of the form (Figure 3):
S ’ ~= I 6,’ exp(-y2/22)
(15)
The maximum retardation 60’ reached a plateau in the limit of high strain rates with a saturation value of -2.7 x rad (or A/2300) for a 100 ppm PS solution in decalin or in toluene and -3.3 x rad in l-methylnaphthalene (Figure 4). When working with a low-viscosity solvent like toluene, some turbulence could be observed at flow rates Q > 50 mumin (gapp > 6000 SKI). In some instances where turbulences did not develop, a perfect Gaussian profile could be obtained even at the highest flow rate of 205 mumin. In a given solvent, the standard deviation u of the retardation profile was found to be dependent on the strain rate and polymer MW. The effect, nevertheless, is weak and amounts to less than 10% in the dilute regime. The width of the retardation signal was similar in decalin and in toluene with u equal to 27-30 pm (jets diameter 500 pm). This value increased in l-methylnaphthalene to 33-37 pm. It should be remembered that the experimental u contains a contribution from the width of the laser beam used t o probe the retardation. Since lf (-17 mm from eq 13) was much larger than the nozzle size, dfcould be taken as constant over the birefringence domain. The real standard deviation, u*, could then be estimated from the convolution relation of two Gaussian distribution functions:
2 = a*2+ of2
(16)
where of= dd4 is the standard deviation of the intensity distribution of the laser beam. The values of u* after correction were 23-26 pm in decalin or toluene and 3034 pm in 1-methylnaphthalene. The inverse Abel transform of a Gaussian function is also a Gaussian function of identical standard deviation:
Birefringence of a PS Solution in Elongational Flow 4855
Macromolecules, Vol. 28, No. 14, 1995 A
r
I
I
I
28’300 ( l i s ) 3 17’000 ( l i s )
I
I
11’300 ( 1 1 1 ) I
0 5’600 ( l i s )
l 0
0
0.002
0001
0
11 : .300
,
, .ZOO
I
I
I
.lo0
0
1
, 100
!
17000 (lis)
I O
11’300 (1,s)
I
I
200
,
~
0.002
I
o !
r -
300
.300
u U
z
!
.ZOO
.lo0
x (rm)
Figure 6. Maximum retardation 6’0 for a PS mixture in decalin (90 ppm 6.85 x lo6 Mp + 90 ppm 4.34 x lo6 Mp) recorded along the flow direction, at different kapp. The orifice entrances are respectively at x = -300 and 300 pm.
m
A 4’200 (lis)
1
0
100
x (rm)
Figure 6. Maximum retardation 6’0 for a PS mixture in 1-methylr )hthalene (90 ppm 6.85 x lo6 Mp + 90 ppm 4.34 x lo6 M,) !corded along the flow direction, at different kapp. 6.OE-3
I
I
Obviously, the difference in the width of the birefringence zone is reflected in the retardation signal. Since o is 1.22 times larger in 1-methylnaphthalene than in decalin and toluene, the retardation signal should also be higher by the same factor in that solvent. This result has indeed been verified as is shown in Figure 4. The maximum retardations, recorded at different positions along the symmetry axis, are given in Figures 5 and 6. Different behaviors were noted, depending on the values of $. At low strain rates, &’(XI was almost constant along the flow direction ( x ) , indicating similar degrees of segmental orientation (Figure 5). At higher strain rates, an increase in the retardation signal was detected in the direction of the jets. This effect was particularly visible in 1-methylnaphthalene where the change in do’ reached almost 100% at &app = 28 300 s-l (Figures 6 and 7). The retardation profiles recorded near the orifices remained Gaussian with
